C Second-order fields and acoustic streaming
As described by eqn (9) the non-linear interactions of the firstorder
fields give rise to time-averaged second-order fields,
resulting in a static pressure field and a steady velocity field as
shown in Fig. 7 . The time-averaged second-order fields exhibit a
spatial oscillation in the horizontal y-direction of wave length
l/2. This is reminiscent of the spatial period doubling for the
classical infinite parallel-plate system of Fig. 1 .
The magnitude of the time-averaged second-order pressure
Sp2T in Fig. 7(a) is approximately 2.5 6 1025 times smaller than
the amplitude of the oscillating first-order pressure in Fig. 5(a).
The time-averaged velocity field Sv2T in Fig. 7(b) contains four
bulk (Rayleigh) streaming rolls. The streaming velocity is
maximum close to the top and bottom wall and has two local
maxima on the horizontal center axis z = 0, where opposite pairs
of bulk rolls meet. The solution in Fig. 7(b) is similar to
Rayleigh’s analytical solution17,20 sketched in Fig. 1, but it
deviates on the following two points: (i) as we consider a
rectangular geometry and not parallel plates, the velocity is
forced to be zero at the side walls, which slightly slows down the
rotational flow of the streaming rolls, and (ii) as we are not in the
limit h % l, the strength of the streaming rolls decreases slightly
before meeting in the center of the channel, which results in a
lower magnitude of the streaming velocity in the horizontal
center plane than predicted by Rayleigh.
Fig. 7(c) shows a zoom-in on the 0.4-mm-thick viscous
boundary layer close to the bottom wall containing two
boundary (Schlicting) streaming rolls. These boundary rolls are
very elongated in the horizontal direction; the z-axis in Fig. 7(c)
is stretched nearly a factor 103 relative to the y-axis. It is
important to mention that the boundary streaming rolls are
generated by the non-linear interactions of the first-order fields
inside the boundary layer, whereas the bulk streaming rolls are
driven by the boundary streaming rolls and not by the non-linear
interaction of the first-order fields in the bulk. The time-averaged
second-order velocity Sv2T is zero at the bottom wall, thus
fulfilling the boundary conditions eqn (4b), while the maximum
of its horizontal component v2y
bnd = 6.42 mm s21 is reached at a
distance of approximately 3d from the wall. The maximum bulk
amplitude U1 = 0.162 m s21 of the horizontal first-order velocity
component, v1y shown in Fig. 5(c), is reached at the channel
center y = 0. From this we calculate the characteristic velocity
ratio Y = c0 vbnd
2y /U1
2 = 0.367, which deviates less than 3% from
the value Y = 3/8 = 0.375 of the parallel-plate model eqn (10).
D Particle tracing simulations
In most experimental microfluidic flow visualization techniques,
tracer particles are employed.42 To mimic this and to ease
comparison with experiment, we have performed particle tracing
simulations using the technique described in Section III B. In all
simulations, we have studied the motion of 144 polystyrene
microparticles suspended in water and distributed evenly at the
initial time t = 0 as shown in Fig. 8(a).
In Fig. 8(b)–(f), the particle trajectories after 10 s of
acoustophoretic motion of the 144 microparticles are shown.
Within each panel, all particles have the same diameter 2a, but
the particle size is progressively enlarged from one panel to the
next: (b) 2a = 0.5 mm, (c) 1 mm, (d) 2 mm, (e) 3 mm, and (f) 5 mm.
For the smallest particles, panels (b) and (c), the drag force from
the acoustic streaming dominates the particle motion, and the
characteristic streaming flow rolls are clearly visualized. For the
larger particles, panels (e) and (f), the acoustic radiation force
dominates the particle motion, and the particle velocity u is nearly
horizontal with the sinusoidal spatial dependence given by uy(y) =
F rad
1D (y)/(6pga) found from eqn (13a). This results in a focusing
motion of the particles towards the vertical pressure nodal plane at
y = 0. Panels (d) and (e) show an intermediate regime where drag
and radiation forces are of the same order of magnitude.
At the nodal plane y = 0 the radiation forces are zero, and
consequently for times t larger than 10 s all particles in panel (f)
that have reached y = 0 end up at (y,z) = (0,¡h/2) due to the
weak but non-zero streaming-induced drag forces.
The cross-over from one acoustophoretic behavior to the
other as a function of particle size, with a critical particle
diameter of 2 mm found in Fig. 8(d), is in agreement with the
following scaling argument:16 If we assume a force balance
between the radiation force and the drag force from acoustic
streaming, Frad = 2Fdrag, keeping a given particle fixed (u = 0),
and if we estimate the magnitude of the streaming velocity to be
given by eqn (10) as Sv2T = YU1
2/c0, where Y is a geometrydependent
factor of order unity, then eqn (13) and (14) lead to
cq r0U2
pa3
1 W&6pgac Y
U2
1
c0
, (19)
where ac is the critical particle radius. Thus, as found in eqn (16),
the critical particle diameter 2ac becomes
2ac~
ffiffiffiffiffiffiffiffiffiffi
12
Y
W
r
d&2:0 mm: (20)
The value is calculated using Y = 0.375, valid for a planar wall
(eqn (10)), and W = 0.165, obtained for polystyrene particles with
diameters between 0.5 mm and 5 mm in water obtained from eqn
(13b) using the parameter values from Table 1. The relation
eqn (20) for the critical cross-over particle diameter is important
for designing experiments relying on specific acoustophoretic
behaviors as function of particle size. Channel geometry enters
through the factor Y, particle and liquid material parameters
through W, and liquid parameters and frequency through the
boundary layer thickness d.
E Streaming for an increased aspect ratio
As an example of how geometry affects the acoustophoretic
motion of polystyrene microparticles, we study here the
consequences of increasing the aspect ratio of the channel
cross-section from h/w = 0.42 to 2 keeping all other parameters
fixed. As illustrated in Fig. 9(a), the streaming velocity field is
only significant close to the top and bottom of the channel for
the large aspect ratio h/w = 2. This happens because given
enough vertical space, the vertical extension D of the streaming
roll is identical to the horizontal one, which is D = l/4. For the
horizontal half-wave resonance in a channel of aspect ratio h/w =
2 we have l = 2w = h, which implies D = h/4, and we therefore
expect a streaming-free region with a vertical extent of h 2 2(h/4)
= h/2 around the center of the channel, which indeed is seen in
Fig. 9(a).
4624 | Lab Chip, 2012, 12, 4617–4627 This journal is The Royal Society of Chemistry 2012
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Published on 23 July 2012 on http://pubs.rsc.org | doi:10.1039/C2LC40612H
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